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The Algebra of Knots
Sam Nelson
Claremont McKenna College
Sam Nelson The Algebra of Knots
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Familiar Operations and Sets
Addition comes from unions:
Sam Nelson The Algebra of Knots
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Familiar Operations and Sets
Addition comes from unions:
Sam Nelson The Algebra of Knots
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Familiar Operations and Sets
Multiplication comes from matched pairs:
Sam Nelson The Algebra of Knots
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Familiar Operations and Sets
Multiplication comes from matched pairs:
Sam Nelson The Algebra of Knots
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Algebraic Properties
The properties of these familiar operations reflect properties ofthe set operations which inspired them.
Sam Nelson The Algebra of Knots
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Algebraic Properties
Example: Associativity of addition
Sam Nelson The Algebra of Knots
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Algebraic Properties
Example: Associativity of addition
Sam Nelson The Algebra of Knots
![Page 9: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/9.jpg)
Algebraic Properties
Example: Associativity of addition
Sam Nelson The Algebra of Knots
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Algebraic Properties
Example: Commutativity of multiplication
Sam Nelson The Algebra of Knots
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Algebraic Properties
Example: Commutativity of multiplication
Sam Nelson The Algebra of Knots
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Algebraic Properties
Example: Commutativity of multiplication
Sam Nelson The Algebra of Knots
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Why sets?
Many useful real-world quantities behave like sets
Examples: lengths, masses, volumes, money
However, not every quantity behaves so simply
Examples: waves interfere, particles become entangled
Let us now consider operations inspired by knots:
Sam Nelson The Algebra of Knots
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Why sets?
Many useful real-world quantities behave like sets
Examples: lengths, masses, volumes, money
However, not every quantity behaves so simply
Examples: waves interfere, particles become entangled
Let us now consider operations inspired by knots:
Sam Nelson The Algebra of Knots
![Page 15: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/15.jpg)
Why sets?
Many useful real-world quantities behave like sets
Examples: lengths, masses, volumes, money
However, not every quantity behaves so simply
Examples: waves interfere, particles become entangled
Let us now consider operations inspired by knots:
Sam Nelson The Algebra of Knots
![Page 16: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/16.jpg)
Why sets?
Many useful real-world quantities behave like sets
Examples: lengths, masses, volumes, money
However, not every quantity behaves so simply
Examples: waves interfere, particles become entangled
Let us now consider operations inspired by knots:
Sam Nelson The Algebra of Knots
![Page 17: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/17.jpg)
Why sets?
Many useful real-world quantities behave like sets
Examples: lengths, masses, volumes, money
However, not every quantity behaves so simply
Examples: waves interfere, particles become entangled
Let us now consider operations inspired by knots:
Sam Nelson The Algebra of Knots
![Page 18: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/18.jpg)
Why sets?
Many useful real-world quantities behave like sets
Examples: lengths, masses, volumes, money
However, not every quantity behaves so simply
Examples: waves interfere, particles become entangled
Let us now consider operations inspired by knots:
Sam Nelson The Algebra of Knots
![Page 19: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/19.jpg)
Knots
Definition: A knot is a simple closed curve inthree-dimensional space.
simple - does not intersect itself
closed - has no loose endpoints, i.e. forms a loop
Sam Nelson The Algebra of Knots
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Knots
Definition: A knot is a simple closed curve inthree-dimensional space.
simple - does not intersect itself
closed - has no loose endpoints, i.e. forms a loop
Sam Nelson The Algebra of Knots
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Knots
Definition: A knot is a simple closed curve inthree-dimensional space.
simple - does not intersect itself
closed - has no loose endpoints, i.e. forms a loop
Sam Nelson The Algebra of Knots
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Knot Diagrams
We represent knots with pictures called knot diagrams.
Sam Nelson The Algebra of Knots
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Knot Diagrams
Drawing a knot from a different angle or moving it around inspace will result in different diagrams of the same knot.
Sam Nelson The Algebra of Knots
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Topology
Knots are topological objects, meaning that we consider twoknots the same if one can be changed into the other by:
moving the knot in space
stretching or shrinking in a continuous way
but not cutting and retying
Sam Nelson The Algebra of Knots
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Topology
Knots are topological objects, meaning that we consider twoknots the same if one can be changed into the other by:
moving the knot in space
stretching or shrinking in a continuous way
but not cutting and retying
Sam Nelson The Algebra of Knots
![Page 26: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/26.jpg)
Topology
Knots are topological objects, meaning that we consider twoknots the same if one can be changed into the other by:
moving the knot in space
stretching or shrinking in a continuous way
but not cutting and retying
Sam Nelson The Algebra of Knots
![Page 27: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/27.jpg)
Topology
Knots are topological objects, meaning that we consider twoknots the same if one can be changed into the other by:
moving the knot in space
stretching or shrinking in a continuous way
but not cutting and retying
Sam Nelson The Algebra of Knots
![Page 28: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/28.jpg)
Why Knots?
Many molecules (polymers, protein, DNA) are knots, andtheir chemical properties are determined in part by howthey’re knotted
Certain antibiotics work by blocking the action ofmolecules called topoisomerase which change how DNA isknotted; blocking the unknotting of the DNA stops thebacteria reproducing
Perhaps surprisingly, the mathematics of knots is relevantto the search for a theory of quantum gravity, a majorunsolved problem in physics
Besides, knots are just fun!
Sam Nelson The Algebra of Knots
![Page 29: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/29.jpg)
Why Knots?
Many molecules (polymers, protein, DNA) are knots, andtheir chemical properties are determined in part by howthey’re knotted
Certain antibiotics work by blocking the action ofmolecules called topoisomerase which change how DNA isknotted; blocking the unknotting of the DNA stops thebacteria reproducing
Perhaps surprisingly, the mathematics of knots is relevantto the search for a theory of quantum gravity, a majorunsolved problem in physics
Besides, knots are just fun!
Sam Nelson The Algebra of Knots
![Page 30: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/30.jpg)
Why Knots?
Many molecules (polymers, protein, DNA) are knots, andtheir chemical properties are determined in part by howthey’re knotted
Certain antibiotics work by blocking the action ofmolecules called topoisomerase which change how DNA isknotted; blocking the unknotting of the DNA stops thebacteria reproducing
Perhaps surprisingly, the mathematics of knots is relevantto the search for a theory of quantum gravity, a majorunsolved problem in physics
Besides, knots are just fun!
Sam Nelson The Algebra of Knots
![Page 31: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/31.jpg)
Why Knots?
Many molecules (polymers, protein, DNA) are knots, andtheir chemical properties are determined in part by howthey’re knotted
Certain antibiotics work by blocking the action ofmolecules called topoisomerase which change how DNA isknotted; blocking the unknotting of the DNA stops thebacteria reproducing
Perhaps surprisingly, the mathematics of knots is relevantto the search for a theory of quantum gravity, a majorunsolved problem in physics
Besides, knots are just fun!
Sam Nelson The Algebra of Knots
![Page 32: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/32.jpg)
Why Knots?
Many molecules (polymers, protein, DNA) are knots, andtheir chemical properties are determined in part by howthey’re knotted
Certain antibiotics work by blocking the action ofmolecules called topoisomerase which change how DNA isknotted; blocking the unknotting of the DNA stops thebacteria reproducing
Perhaps surprisingly, the mathematics of knots is relevantto the search for a theory of quantum gravity, a majorunsolved problem in physics
Besides, knots are just fun!
Sam Nelson The Algebra of Knots
![Page 33: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/33.jpg)
The main problem
Given two knot diagrams K and K ′, how can we tell whetherthey represent the same knot?
Sam Nelson The Algebra of Knots
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Reidemeister Moves
Sam Nelson The Algebra of Knots
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Example
Sam Nelson The Algebra of Knots
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Knot Invariants
Quantities we can compute from a knot diagram
Get the same value for all diagrams of the same knot
Unchanged by Reidemeister moves
If K and K ′ have different invariant values, they representdifferent knots
Sam Nelson The Algebra of Knots
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Knot Invariants
Quantities we can compute from a knot diagram
Get the same value for all diagrams of the same knot
Unchanged by Reidemeister moves
If K and K ′ have different invariant values, they representdifferent knots
Sam Nelson The Algebra of Knots
![Page 38: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/38.jpg)
Knot Invariants
Quantities we can compute from a knot diagram
Get the same value for all diagrams of the same knot
Unchanged by Reidemeister moves
If K and K ′ have different invariant values, they representdifferent knots
Sam Nelson The Algebra of Knots
![Page 39: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/39.jpg)
Knot Invariants
Quantities we can compute from a knot diagram
Get the same value for all diagrams of the same knot
Unchanged by Reidemeister moves
If K and K ′ have different invariant values, they representdifferent knots
Sam Nelson The Algebra of Knots
![Page 40: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/40.jpg)
Knot Invariants
Quantities we can compute from a knot diagram
Get the same value for all diagrams of the same knot
Unchanged by Reidemeister moves
If K and K ′ have different invariant values, they representdifferent knots
Sam Nelson The Algebra of Knots
![Page 41: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/41.jpg)
Knot Invariants
Examples:
Alexander/Jones/HOMFLYpt/Kauffman polynomials
Knot group
Hyperbolic Volume
TQFTs
Khovanov Homology
Sam Nelson The Algebra of Knots
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Knot Invariants
Examples:
Alexander/Jones/HOMFLYpt/Kauffman polynomials
Knot group
Hyperbolic Volume
TQFTs
Khovanov Homology
Sam Nelson The Algebra of Knots
![Page 43: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/43.jpg)
Knot Invariants
Examples:
Alexander/Jones/HOMFLYpt/Kauffman polynomials
Knot group
Hyperbolic Volume
TQFTs
Khovanov Homology
Sam Nelson The Algebra of Knots
![Page 44: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/44.jpg)
Knot Invariants
Examples:
Alexander/Jones/HOMFLYpt/Kauffman polynomials
Knot group
Hyperbolic Volume
TQFTs
Khovanov Homology
Sam Nelson The Algebra of Knots
![Page 45: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/45.jpg)
Knot Invariants
Examples:
Alexander/Jones/HOMFLYpt/Kauffman polynomials
Knot group
Hyperbolic Volume
TQFTs
Khovanov Homology
Sam Nelson The Algebra of Knots
![Page 46: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/46.jpg)
Knot Invariants
Examples:
Alexander/Jones/HOMFLYpt/Kauffman polynomials
Knot group
Hyperbolic Volume
TQFTs
Khovanov Homology
Sam Nelson The Algebra of Knots
![Page 47: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/47.jpg)
An algebraic knot invariant
Define an algebraic operation . from knot diagrams
Use Reidemeister moves to determine properties of theoperation
Find operations satisfying these axioms either by combiningexisting operations or by making operation tables
Sam Nelson The Algebra of Knots
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An algebraic knot invariant
Define an algebraic operation . from knot diagrams
Use Reidemeister moves to determine properties of theoperation
Find operations satisfying these axioms either by combiningexisting operations or by making operation tables
Sam Nelson The Algebra of Knots
![Page 49: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/49.jpg)
An algebraic knot invariant
Define an algebraic operation . from knot diagrams
Use Reidemeister moves to determine properties of theoperation
Find operations satisfying these axioms either by combiningexisting operations or by making operation tables
Sam Nelson The Algebra of Knots
![Page 50: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/50.jpg)
An algebraic knot invariant
Define an algebraic operation . from knot diagrams
Use Reidemeister moves to determine properties of theoperation
Find operations satisfying these axioms either by combiningexisting operations or by making operation tables
Sam Nelson The Algebra of Knots
![Page 51: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/51.jpg)
Kei
(a.k.a. Involutory Quandles)
Attach a label to each arc in a knot diagram
When x goes under y, the result is x . y
Sam Nelson The Algebra of Knots
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Kei
(a.k.a. Involutory Quandles)
Attach a label to each arc in a knot diagram
When x goes under y, the result is x . y
Sam Nelson The Algebra of Knots
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Kei
(a.k.a. Involutory Quandles)
Attach a label to each arc in a knot diagram
When x goes under y, the result is x . y
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Kei
(a.k.a. Involutory Quandles)
Attach a label to each arc in a knot diagram
When x goes under y, the result is x . y
Sam Nelson The Algebra of Knots
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Kei Axioms
x . x = x
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Kei Axioms
(x . y) . y = x
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Kei Axioms
(x . y) . z = (x . z) . (y . z)
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Kei
Thus, a kei is a set of labels with an operation . such that forall x, y, z we have
(i) x . x = x
(ii) (x . y) . y = x
(iii) (x . y) . z = (x . z) . (y . z)
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Kei
Thus, a kei is a set of labels with an operation . such that forall x, y, z we have
(i) x . x = x
(ii) (x . y) . y = x
(iii) (x . y) . z = (x . z) . (y . z)
Sam Nelson The Algebra of Knots
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Kei
Thus, a kei is a set of labels with an operation . such that forall x, y, z we have
(i) x . x = x
(ii) (x . y) . y = x
(iii) (x . y) . z = (x . z) . (y . z)
Sam Nelson The Algebra of Knots
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Kei
Thus, a kei is a set of labels with an operation . such that forall x, y, z we have
(i) x . x = x
(ii) (x . y) . y = x
(iii) (x . y) . z = (x . z) . (y . z)
Sam Nelson The Algebra of Knots
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Kei Example
We can define kei operations using existing operations. Forexample, the integers Z have a kei operation given by
x . y = 2y − x.
To see that this is a kei operation, we need to verify that itsatisfies the axioms. For example,
x . x = 2x− x = x
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Kei Example
We can define kei operations using existing operations. Forexample, the integers Z have a kei operation given by
x . y = 2y − x.
To see that this is a kei operation, we need to verify that itsatisfies the axioms. For example,
x . x = 2x− x = x
Sam Nelson The Algebra of Knots
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Kei Example
We can define kei operations using existing operations. Forexample, the integers Z have a kei operation given by
x . y = 2y − x.
To see that this is a kei operation, we need to verify that itsatisfies the axioms. For example,
x . x = 2x− x = x
Sam Nelson The Algebra of Knots
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Kei Example
We can also specify a kei operation with an operationtable, just like a multiplication table:
. 1 2 3
1 1 3 22 3 2 13 2 1 3
Checking the axioms here must be done case-by-case and isbest handled by computer code.
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Kei Example
We can also specify a kei operation with an operationtable, just like a multiplication table:
. 1 2 3
1 1 3 22 3 2 13 2 1 3
Checking the axioms here must be done case-by-case and isbest handled by computer code.
Sam Nelson The Algebra of Knots
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Kei Example
We can also specify a kei operation with an operationtable, just like a multiplication table:
. 1 2 3
1 1 3 22 3 2 13 2 1 3
Checking the axioms here must be done case-by-case and isbest handled by computer code.
Sam Nelson The Algebra of Knots
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The counting invariant
A valid labeling of a knot diagram by a kei must satisfy thecrossing condition at every crossing.
. 1 2 3
1 1 3 22 3 2 13 2 1 3
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The counting invariant
If a labeling of a knot diagram by a kei fails to satisfy thecrossing condition at any crossing, it is invalid.
. 1 2 3
1 1 3 22 3 2 13 2 1 3
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The counting invariant
Because of the kei axioms, every valid labeling of a diagrambefore a move corresponds to a unique valid labeling after amove.
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The counting invariant
Any two diagrams of the same knot will have same number ofvalid labelings by your favorite kei. So, to tell knots apart,
we count the valid labelings!
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The counting invariant
Any two diagrams of the same knot will have same number ofvalid labelings by your favorite kei. So, to tell knots apart,
we count the valid labelings!
Sam Nelson The Algebra of Knots
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The counting invariant
If our set of labels is finite, there are only finitely manypossible labelings
So we can list all possible labelings and count how manyare valid
This number will be the same for all diagrams of K
So it is a knot invariant, called the kei counting invariant
Sam Nelson The Algebra of Knots
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The counting invariant
If our set of labels is finite, there are only finitely manypossible labelings
So we can list all possible labelings and count how manyare valid
This number will be the same for all diagrams of K
So it is a knot invariant, called the kei counting invariant
Sam Nelson The Algebra of Knots
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The counting invariant
If our set of labels is finite, there are only finitely manypossible labelings
So we can list all possible labelings and count how manyare valid
This number will be the same for all diagrams of K
So it is a knot invariant, called the kei counting invariant
Sam Nelson The Algebra of Knots
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The counting invariant
If our set of labels is finite, there are only finitely manypossible labelings
So we can list all possible labelings and count how manyare valid
This number will be the same for all diagrams of K
So it is a knot invariant, called the kei counting invariant
Sam Nelson The Algebra of Knots
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The counting invariant
If our set of labels is finite, there are only finitely manypossible labelings
So we can list all possible labelings and count how manyare valid
This number will be the same for all diagrams of K
So it is a knot invariant, called the kei counting invariant
Sam Nelson The Algebra of Knots
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The counting invariant
The unknot has three labelings by our three-element kei:
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A knot invariant
The trefoil has nine labelings by our three-element kei:
Sam Nelson The Algebra of Knots
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A knot invariant
3 6= 9, and the kei counting invariant shows that no sequence ofReidemeister moves can unknot the trefoil.
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Generalizations of Kei
An oriented knot has a preferred direction of travel.
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Generalizations of Kei
Orienting K changes axiom (ii) to allow .−1 to be differentfrom .; the resulting object is called a quandle
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Generalizations of Kei
Orienting K changes axiom (ii) to allow .−1 to be differentfrom .; the resulting object is called a quandle
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Generalizations of Kei
Replacing the type I move with a doubled version gives usframed knots which are like physical knots
This eliminates quandle axiom (i); the resulting object iscalled a rack
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Generalizations of Kei
Replacing the type I move with a doubled version gives usframed knots which are like physical knots
This eliminates quandle axiom (i); the resulting object iscalled a rack
Sam Nelson The Algebra of Knots
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Generalizations of Kei
Replacing the type I move with a doubled version gives usframed knots which are like physical knots
This eliminates quandle axiom (i); the resulting object iscalled a rack
Sam Nelson The Algebra of Knots
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Generalizations of Kei
Dividing the arcs at over-crossings to get semi-arcs letsboth inputs act on each other
The resulting algebraic structures are called bikei,biquandles and biracks
These are solutions to the Yang-Baxter Equation fromstatistical mechanics
Sam Nelson The Algebra of Knots
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Generalizations of Kei
Dividing the arcs at over-crossings to get semi-arcs letsboth inputs act on each other
The resulting algebraic structures are called bikei,biquandles and biracks
These are solutions to the Yang-Baxter Equation fromstatistical mechanics
Sam Nelson The Algebra of Knots
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Generalizations of Kei
Dividing the arcs at over-crossings to get semi-arcs letsboth inputs act on each other
The resulting algebraic structures are called bikei,biquandles and biracks
These are solutions to the Yang-Baxter Equation fromstatistical mechanics
Sam Nelson The Algebra of Knots
![Page 90: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/90.jpg)
Generalizations of Kei
Dividing the arcs at over-crossings to get semi-arcs letsboth inputs act on each other
The resulting algebraic structures are called bikei,biquandles and biracks
These are solutions to the Yang-Baxter Equation fromstatistical mechanics
Sam Nelson The Algebra of Knots
![Page 91: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/91.jpg)
Enhanced Invariants
A signature is a quantity computable from a labeled knotdiagram which is invariant under labeled moves
Counting signatures instead of labelings defines strongerinvariants
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Enhanced Invariants
A signature is a quantity computable from a labeled knotdiagram which is invariant under labeled moves
Counting signatures instead of labelings defines strongerinvariants
Sam Nelson The Algebra of Knots
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Enhanced Invariants
A signature is a quantity computable from a labeled knotdiagram which is invariant under labeled moves
Counting signatures instead of labelings defines strongerinvariants
Sam Nelson The Algebra of Knots
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Enhanced Invariants
Example: For each labeling, count uc where c is the number oflabels used. This is called the image-enhanced countinginvariant, denoted ΦIm
X .
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Enhanced Invariants
ΦImX (Unknot) = u + u + u = 3u
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Enhanced Invariants
ΦImX (31) = 3u + 6u3
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Enhancements
Enhancements can come from the general structure of thelabeling object.
Image enhancements
Writhe enhancements
Quandle polynomials
Column groups
Sam Nelson The Algebra of Knots
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Enhancements
Enhancements can come from the general structure of thelabeling object.
Image enhancements
Writhe enhancements
Quandle polynomials
Column groups
Sam Nelson The Algebra of Knots
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Enhancements
Enhancements can come from the general structure of thelabeling object.
Image enhancements
Writhe enhancements
Quandle polynomials
Column groups
Sam Nelson The Algebra of Knots
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Enhancements
Enhancements can come from the general structure of thelabeling object.
Image enhancements
Writhe enhancements
Quandle polynomials
Column groups
Sam Nelson The Algebra of Knots
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Enhancements
Enhancements can come from the general structure of thelabeling object.
Image enhancements
Writhe enhancements
Quandle polynomials
Column groups
Sam Nelson The Algebra of Knots
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Enhancements
Enhancements can come from using special types of labelingobjects.
Symplectic quandle invariants
Bilinear biquandle invariants
Coxeter rack invariants
(t, s)-rack invariants
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![Page 103: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/103.jpg)
Enhancements
Enhancements can come from using special types of labelingobjects.
Symplectic quandle invariants
Bilinear biquandle invariants
Coxeter rack invariants
(t, s)-rack invariants
Sam Nelson The Algebra of Knots
![Page 104: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/104.jpg)
Enhancements
Enhancements can come from using special types of labelingobjects.
Symplectic quandle invariants
Bilinear biquandle invariants
Coxeter rack invariants
(t, s)-rack invariants
Sam Nelson The Algebra of Knots
![Page 105: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/105.jpg)
Enhancements
Enhancements can come from using special types of labelingobjects.
Symplectic quandle invariants
Bilinear biquandle invariants
Coxeter rack invariants
(t, s)-rack invariants
Sam Nelson The Algebra of Knots
![Page 106: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/106.jpg)
Enhancements
Enhancements can come from using special types of labelingobjects.
Symplectic quandle invariants
Bilinear biquandle invariants
Coxeter rack invariants
(t, s)-rack invariants
Sam Nelson The Algebra of Knots
![Page 107: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/107.jpg)
Enhancements
Enhancements can also come from additional structures.
Quandle 2-cocycles
Rack shadows
Rack algebras
Sam Nelson The Algebra of Knots
![Page 108: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/108.jpg)
Enhancements
Enhancements can also come from additional structures.
Quandle 2-cocycles
Rack shadows
Rack algebras
Sam Nelson The Algebra of Knots
![Page 109: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/109.jpg)
Enhancements
Enhancements can also come from additional structures.
Quandle 2-cocycles
Rack shadows
Rack algebras
Sam Nelson The Algebra of Knots
![Page 110: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/110.jpg)
Enhancements
Enhancements can also come from additional structures.
Quandle 2-cocycles
Rack shadows
Rack algebras
Sam Nelson The Algebra of Knots
![Page 111: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/111.jpg)
Tying it all up...
Algebra from knots seems weird at first,
but the algebraic structures arising from knots haveapplication in biology, chemistry, physics. . .
and who knows where else?
You can help find out!
Sam Nelson The Algebra of Knots
![Page 112: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/112.jpg)
Tying it all up...
Algebra from knots seems weird at first,
but the algebraic structures arising from knots haveapplication in biology, chemistry, physics. . .
and who knows where else?
You can help find out!
Sam Nelson The Algebra of Knots
![Page 113: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/113.jpg)
Tying it all up...
Algebra from knots seems weird at first,
but the algebraic structures arising from knots haveapplication in biology, chemistry, physics. . .
and who knows where else?
You can help find out!
Sam Nelson The Algebra of Knots
![Page 114: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/114.jpg)
Tying it all up...
Algebra from knots seems weird at first,
but the algebraic structures arising from knots haveapplication in biology, chemistry, physics. . .
and who knows where else?
You can help find out!
Sam Nelson The Algebra of Knots
![Page 115: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/115.jpg)
Tying it all up...
Algebra from knots seems weird at first,
but the algebraic structures arising from knots haveapplication in biology, chemistry, physics. . .
and who knows where else?
You can help find out!
Sam Nelson The Algebra of Knots
![Page 116: The Algebra of Knots - Fullerton CollegestaffSam Nelson The Algebra of Knots. Why Knots? Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined](https://reader030.vdocuments.net/reader030/viewer/2022021716/5e6e73f8e7cb645aca67c89b/html5/thumbnails/116.jpg)
Thanks for Listening!
Sam Nelson The Algebra of Knots